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Rigorous computation of high-order aberrations and their errors.
1Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA.
Microscopy (Oxford, England)
|April 4, 2026
Summary
High-order transfer maps provide valuable insights into optical systems. New methods rigorously bound missing orders, ensuring map accuracy and enabling precise step-size control in Differential Algebra integration.
Area of Science:
- Physics
- Applied Mathematics
- Optical Engineering
Background:
- High-order transfer maps are crucial for analyzing single-pass and multipass optical systems.
- These maps compute properties like dispersions, chromaticities, and tune shifts.
- Assessing the accuracy of a given map's expansion order is a persistent challenge.
Purpose of the Study:
- To develop methods for determining mathematically rigorous bounds on all missing orders of high-order transfer maps.
- To address the question of map accuracy beyond the explicitly computed expansion order.
- To enable rigorous step-size control in Differential Algebra (DA)-based integration.
Main Methods:
- Computation of high-order transfer maps.
- Development of algorithms to derive rigorous bounds on neglected higher-order terms.
- Application of Differential Algebra (DA) for integration and step-size control.
Main Results:
- Established methods to mathematically bound all missing orders of transfer maps.
- Demonstrated that the computational cost for bounds is significantly less than map computation.
- Enabled rigorous step-size control in DA-based integration of high-order maps.
Conclusions:
- The presented methods provide a rigorous way to assess the accuracy of high-order transfer maps.
- These techniques enhance the reliability of optical system analysis using transfer maps.
- The approach facilitates more precise and controlled computations in optical system design.
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